Problem 1

Prove that for nonnegative \(a, b, c\) the following inequality holds: \[\left(\frac{a(b+c)}{4a+b+c} \right)^3+\left(\frac{b(c+a)}{4b+c+a} \right)^3+ \left(\frac{c(a+b)}{4c+a+b} \right)^3 \leq \frac{(a+b)(b+c)(c+a)}{72}\]

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Tags: inequality

Solution

Denote \(x=\frac{b+c}{2}, y=\frac{c+a}{2}, z=\frac{a+b}{2}\). Note that \[xyz \geq (-x+y+z)(x-y+z)(x+y-z)=\sum_{cyc} x \cdot x \cdot (-x+y+z)-2xyz \geq \sum_{cyc} \left(\frac{3}{\frac{1}{x}+\frac{1}{x}+\frac{1}{-x+y+z}}\right)^3-2xyz.\] Now it remains to substitute the variables.